Finite Group Databasebeta

← #34099252

Subgroups — #34099252

Order 20096.

314 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757645C211yesmaximal, characteristic, complemented
2#23757563C212yesmaximal, complemented
3#23757017C211yesmaximal, characteristic, complemented
4D4⋊#11929434C212yesmaximal, complemented
5D42×C157C211yesmaximal, characteristic
6C2×#14578942—22no—
7#19097771C2211yescharacteristic, complemented
8#19097702—22no—
9C4⋊#11929434C2212yescomplemented
10#19097692—22no—
11C4×#11929434C2211yescharacteristic, complemented
12C22×D4×C157—22no—
13#127×C157C2211yescharacteristic
14#121×C157—22no—
15#120×C157—22no—
16C4×D4×C157C2212yes—
17C2×Dic314—21no—
18#14578942—44no—
19C2×#11929434—21no—
20C2×#11929434—22no—
21C4⋊Dic157—22no—
22C24×C157—22no—
23C2×D4×C157—24no—
24C2×D4×C157—22no—
25C2×D4×C157—44no—
26C2×D4×C157—44no—
27C2×D4×C157D412yes—
28C2×D4×C157D412yes—
29C22×C4×C157—22no—
30C4⋊C4×C157D412yes—
31C22⋊C4×C157—22no—
32C22⋊C4×C157—22no—
33C42×C157C2311yescharacteristic
34Dic314—42no—
35C2×Dic157—41no—
36#11929434—22no—
37#11929434—42no—
38C23×C157—22no—
39C23×C157—22no—
40C23×C157—48no—
41C23×C157—82no—
42D4×C157—22no—
43D4×C157—42no—
44D4×C157—48no—
45D4×C157—42no—
46D4×C157—42no—
47D4×C157SD1614yescomplemented
48C2×C4×C157—22no—
49C2×C4×C157—42no—
50C2×C4×C157—42no—
51C2×C4×C157C2×D411yescharacteristic
52C2×C4×C157C2×D412yes—
53C22×C157—24no—
54C22×C157—42no—
55C22×C157—42no—
56C22×C157—44no—
57C22×C157—42no—
58C22×C157—88no—
59C22×C157#12011yescharacteristic
60C4×C157—21no—
61C4×C157—42no—
62C4×C157C2×SD1612yes—
63C4×C157AGL(1,ℤ/8)12yes—
64Dic157—81no—
65C2×C157—44no—
66C2×C157—82no—
67C2×C157#45012yes—
68C2×C157#45311yescharacteristic
69C157#162911yescharacteristic, complemented, Sylow
70#1629—1571nomaximal, Sylow
71D42D15711yescharacteristic
72#492—1571no—
73#476—1572no—
74#331—1571no—
75D4⋊C8—1572no—
76C22×D4—22no—
77C2×SD16—3142no—
78C4⋊Q8—1571no—
79#127C2×D15711yescharacteristic
80#121—22no—
81#120—22no—
82C4×D4C2×D15712yes—
83#106—3142no—
84C4⋊C8—1572no—
85#102—3142no—
86C4×C8—1571no—
87C24—22no—
88C2×Q8—3141no—
89C2×D4—24no—
90C2×D4—22no—
91C2×D4—44no—
92C2×D4—44no—
93C2×D4C22⋊D15712yes—
94C2×D4C22⋊D15712yes—
95C22×C4—22no—
96SD16—6284no—
97C2×C8—3141no—
98C2×C8—3142no—
99C4⋊C4—3142no—
100C4⋊C4C22⋊D15712yes—
101C22⋊C4—22no—
102C22⋊C4—22no—
103C42C22×D15711yescharacteristic
104C23—22no—
105C23—22no—
106C23—48no—
107C23—82no—
108Q8—6282no—
109D4—22no—
110D4—42no—
111D4—48no—
112D4—42no—
113D4—42no—
114D4#1457894214yescomplemented
115C2×C4—22no—
116C2×C4—42no—
117C2×C4—42no—
118C2×C4—6281no—
119C2×C4C2×C22⋊D15711yescharacteristic
120C2×C4C2×C22⋊D15712yes—
121C8—3142no—
122C8—6282no—
123C22—24no—
124C22—42no—
125C22—42no—
126C22—44no—
127C22—42no—
128C22—88no—
129C22#1909774011yescharacteristic
130C4—21no—
131C4—42no—
132C4—12561no—
133C4C2×#1457894212yes—
134C4#1909779712yes—
135C2—44no—
136C2—82no—
137C2#2375742412yes—
138C2#2375744111yescharacteristic
subgroupG/H
centerC22#19097740
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
Fitting, prime radicalD42×C157C2
socleC22×C157#120
hypercenterD42D157
hypocenterC157#1629
LCSG ▷ C42×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ D42

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #23757645C2semidirect
2. #23757563C2semidirect
3. #23757017C2semidirect
4. D4⋊#11929434C2semidirect
7. #19097771C22semidirect
9. C4⋊#11929434C22semidirect
11. C4×#11929434C22semidirect
47. D4×C157SD16semidirect
69. C157#1629semidirect
114. D4#14578942semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#237576452
#237575632
#237570172
D4⋊#119294342
D42×C1572
#1629157